BBS Seminar, 1 November 2019, Basel

Probabilistic DAGs - Bayesian networks

Compact representation of multivariate probability distributions

Probabilistic graphical models for a set of variables \(\{ X_1, X_2, \ldots, X_n \}\) characterized by

  • a graphical structure, directed and acyclic, whose nodes are the variables
  • a probability model for each node describing the relationship with its parents
  • edges encode conditional independencies (any variable is conditionally independent of its non-descendant given its parents)

$P(X_1)P(X_2)P(X_3 \vert X_1, X_2)P(X_4 \vert X_2, X_3)$
e.g. $X_4 \perp\!\!\!\perp X_1 \vert (X_2, X_3)$

\(\{ X_1, X_2, \ldots, X_n \} \thicksim P(X_1, X_2, \ldots, X_n) = \prod_{i=1}^n P(X_i \vert {\textbf{Pa}_i})\)

Thank you



Essential references

  • Moffa, Giusi, et al. “Using directed acyclic graphs in epidemiological research in psychosis: an analysis of the role of bullying in psychosis.” Schizophrenia Bulletin 43.6 (2017): 1273-1279.
  • Jack Kuipers*, Giusi Moffa*, Elizabeth Kuipers, Daniel Freeman and Paul Bebbington. “Links between psychotic and neurotic symptoms in the general population: an analysis of longitudinal British National Survey data using Directed Acyclic Graphs.” Psychological Medicine (2018): 1-8.
  • Kuipers, Jack, and Giusi Moffa. “Partition MCMC for inference on acyclic digraphs.” Journal of the American Statistical Association 112.517 (2017): 282-299.
  • Kuipers, Jack, Polina Suter, and Giusi Moffa. “Efficient Structure Learning and Sampling of Bayesian Networks.” arXiv preprint arXiv:1803.07859 (2018).
  • Kuipers, Jack, Thomas Thurnherr, Giusi Moffa, et al. “Mutational interactions define novel cancer subgroups.” Nature Communications 9, (2018).
  • Dawid, A. Philip. “Beware of the DAG!.” Causality: Objectives and Assessment. 2010.
  • Pearl, Judea. Causality. Cambridge university press, 2009.

Companion slides

Dynamic Bayesian network - graph

2000 British National Psychiatric Morbidity survey and its 18-month follow-up data (N=2406)

  • one node for each variable at each time slice
  • assume stationarity over time
  • edges only displayed if they appear in at least 10% of the sampled DAGs

sample of 10,000 DAGs

Kuipers, Moffa et al, Psych Med 2018